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Theorem frgrncvvdeqlem1 29285
Description: Lemma 1 for frgrncvvdeq 29295. (Contributed by Alexander van der Vekens, 23-Dec-2017.) (Revised by AV, 8-May-2021.) (Proof shortened by AV, 12-Feb-2022.)
Hypotheses
Ref Expression
frgrncvvdeq.v1 𝑉 = (Vtx‘𝐺)
frgrncvvdeq.e 𝐸 = (Edg‘𝐺)
frgrncvvdeq.nx 𝐷 = (𝐺 NeighbVtx 𝑋)
frgrncvvdeq.ny 𝑁 = (𝐺 NeighbVtx 𝑌)
frgrncvvdeq.x (𝜑𝑋𝑉)
frgrncvvdeq.y (𝜑𝑌𝑉)
frgrncvvdeq.ne (𝜑𝑋𝑌)
frgrncvvdeq.xy (𝜑𝑌𝐷)
frgrncvvdeq.f (𝜑𝐺 ∈ FriendGraph )
frgrncvvdeq.a 𝐴 = (𝑥𝐷 ↦ (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸))
Assertion
Ref Expression
frgrncvvdeqlem1 (𝜑𝑋𝑁)

Proof of Theorem frgrncvvdeqlem1
StepHypRef Expression
1 frgrncvvdeq.xy . . . 4 (𝜑𝑌𝐷)
2 df-nel 3051 . . . . 5 (𝑌𝐷 ↔ ¬ 𝑌𝐷)
3 frgrncvvdeq.nx . . . . . 6 𝐷 = (𝐺 NeighbVtx 𝑋)
43eleq2i 2830 . . . . 5 (𝑌𝐷𝑌 ∈ (𝐺 NeighbVtx 𝑋))
52, 4xchbinx 334 . . . 4 (𝑌𝐷 ↔ ¬ 𝑌 ∈ (𝐺 NeighbVtx 𝑋))
61, 5sylib 217 . . 3 (𝜑 → ¬ 𝑌 ∈ (𝐺 NeighbVtx 𝑋))
7 nbgrsym 28353 . . 3 (𝑋 ∈ (𝐺 NeighbVtx 𝑌) ↔ 𝑌 ∈ (𝐺 NeighbVtx 𝑋))
86, 7sylnibr 329 . 2 (𝜑 → ¬ 𝑋 ∈ (𝐺 NeighbVtx 𝑌))
9 frgrncvvdeq.ny . . . 4 𝑁 = (𝐺 NeighbVtx 𝑌)
10 neleq2 3056 . . . 4 (𝑁 = (𝐺 NeighbVtx 𝑌) → (𝑋𝑁𝑋 ∉ (𝐺 NeighbVtx 𝑌)))
119, 10ax-mp 5 . . 3 (𝑋𝑁𝑋 ∉ (𝐺 NeighbVtx 𝑌))
12 df-nel 3051 . . 3 (𝑋 ∉ (𝐺 NeighbVtx 𝑌) ↔ ¬ 𝑋 ∈ (𝐺 NeighbVtx 𝑌))
1311, 12bitri 275 . 2 (𝑋𝑁 ↔ ¬ 𝑋 ∈ (𝐺 NeighbVtx 𝑌))
148, 13sylibr 233 1 (𝜑𝑋𝑁)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205   = wceq 1542  wcel 2107  wne 2944  wnel 3050  {cpr 4593  cmpt 5193  cfv 6501  crio 7317  (class class class)co 7362  Vtxcvtx 27989  Edgcedg 28040   NeighbVtx cnbgr 28322   FriendGraph cfrgr 29244
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-1st 7926  df-2nd 7927  df-nbgr 28323
This theorem is referenced by:  frgrncvvdeqlem7  29291  frgrncvvdeqlem8  29292  frgrncvvdeqlem9  29293
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